Magnetic resonance imaging (MRI) has been used with a variety of specimens in clinical and other applications. In many examples, images are based on differences in the so-called T1 and T2 relaxation times in the specimen being imaged. In other examples, translational diffusion can be used as a relaxation mechanism that produces image contrast. While in some specimens, translational diffusion of spins is isotropic, in many important applications, specimens exhibit structural features that make translational diffusion anisotropic. To evaluate diffusion anisotropies of a specimen, so-called diffusion tensor (DT) methods such as those described in Basser et al., U.S. Pat. No. 5,539,310, can be used. These methods typically involve the application of pulsed-gradient magnetic field sequences in a multiplicity of directions. DT methods are generally based on 3-dimensional Gaussian spin-displacement profiles appropriate at small q-values, wherein q is proportional to a magnitude and a duration of an applied pulsed magnetic-field gradient (“PFG”) MR sequences. DT methods can fail to reveal important specimen properties when the 3-dimensional Gaussian displacement model is not applicable, and improved MR methods and apparatus are needed.